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An integral equation is constructed relating the three-parameter Mittag-Leffler function with a perturbed one, from which a new property is derived. The latter helps in finding new sums of series of Mittag-Leffler functions. Also, a bound is obtained for the Mittag-Leffler function, allowing to perform a sensitivity analysis, being vital in the solution of fractional differential equations. The bound is also tested on a numerical example. [ABSTRACT FROM AUTHOR]

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay. [ABSTRACT FROM AUTHOR]

We study the Volterra integral equation of the first kind with an integral operator of order , a singularity, and a sufficiently smooth kernel in a certain Banach space with weight. It reduces to an integro-differential equation with two terms in the left-hand side. The first term corresponds to an equation for which an explicitly multiparameter family of solutions is constructed. For the second term we obtain an equation with an operator whose norm in an arbitrary Banach space is arbitrarily small near zero. Such splitting of the integral operator allows constructing a particular and general solutions to the integro-differential equations in the corresponding Banach space in the form of convergent series. Thus, under certain restrictions on the operator pencil corresponding to a given integral operator, a multiparameter family of solutions is constructed for the original integral equation. [ABSTRACT FROM AUTHOR]

The objective of this research is to present new fixed point theorems for two separate families of fuzzy-dominated mappings. These mappings must satisfy a unique locally contraction in a complete b-multiplicative metric space. Also, we have obtained novel results for families of fuzzy-dominated mappings on a closed ball that meet the requirements of a generalized locally contraction. This research introduces new and challenging fixed-point problems for families of ordered fuzzy-dominated mappings in ordered complete b-multiplicative metric spaces. Moreover, we demonstrate a new concept for families of fuzzy graph-dominated mappings on a closed ball in these spaces. Additionally, we present novel findings for graphic contraction endowed with graphic structure. These findings are groundbreaking and provide a strong foundation for future research in this field. To demonstrate the uniqueness of our novel findings, we provide evidence of their applicability in obtaining the common solution of integral and fractional differential equations. Our findings have resulted in modifications to several contemporary and classical results in the research literature. This provides further evidence of the originality and impact of our work. [ABSTRACT FROM AUTHOR]

This article studies a inverse problem for diffusion equation on the rectangular domain. First we introduce a definition of a classical solution, and then the original problem investigate to use successive approximation. Second, the given problem is reduced to an equivalent problem. Existence and uniqueness of the solution of the equivalent problem is proved using a contraction mapping principle. Finally, using the equivalency, the existence and uniqueness of classical solution is obtained. [ABSTRACT FROM AUTHOR]

The problem of frictional thermomechanical contact of a rigid cylindrical flat-based punch with an elastic half-space has been solved numerically. The partial slipping with friction between the coupled surfaces, as a result of a compressive load, at the first stage, and a uniform heating of the bodies, at the second stage, has been investigated. To take into account the friction, we have used Coulomb's law. This problem is reduced to systems of nonlinear boundary integral equations that correspond to the mechanical and the thermal stress stages. The discretization of integral equations and iterative process have been used to obtain a numerical solution of these systems. The dependences of the normal and tangential contact stress distributions and the configuration of the stick and slip zones on the temperature have been analyzed for two possible cases of the relationship between the coefficients of thermal expansion of the bodies. [ABSTRACT FROM AUTHOR]

The present work is devoted to finding numerical solutions of two types of nonlinear integral equations on half line with kernels depending on the sum and difference of arguments. These equations arise in various fields of mathematical physics: kinetic theory of gases, theoretical astrophysics, p-adic string theory, etc. The main result of the work is the derivation of an uniform estimate of the norm of difference between two successive approximations of solutions, which plays an important role for the control of the convergence of iterative schemes and number of iterations. The obtained results have been applied to determine numerical solutions of models from different areas of applications. [ABSTRACT FROM AUTHOR]

In this article, two dimensional inverse problem of determining convolution kernel in the fractional diffusion equation with the time-fractional Caputo derivative is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation with Riemann–Liouville derivative is constructed. Using the formulas of asymptotic expansions for the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function, which was used for studying the inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. The local existence and global uniqueness results are proven by the aid of fixed point argument in suitable functional classes. Also the stability estimate is obtained. [ABSTRACT FROM AUTHOR]

The paper considers two weakly singular Fredholm boundary integral equations of the first kind to each of which the three-dimensional Helmholtz transmission problem can be reduced. The properties of these equations are studied on the spectra, where they are ill posed. For the first equation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of the transmission problem. The second equation in this case always has infinitely many solutions, with only one of them giving a solution of the transmission problem. The interpolation method for finding approximate solutions of the integral equations and the transmission problem in question is discussed. [ABSTRACT FROM AUTHOR]

In the present work, an initial-boundary value problem for a high even order partial differential equation with a Bessel operator is considered and the existence, uniqueness and stability of the solution of the considered problem have been investigated. [ABSTRACT FROM AUTHOR]

This work investigates an initial-boundary value and an inverse coefficient problem of determining a space dependent coefficient in the fractional wave equation with the generalized Riemann–Liouville (Hilfer) time derivative. In the beginning, it is considered the initial boundary value problem (direct problem). By the Fourier method, this problem is reduced to equivalent integral equations, which contain Mittag-Leffler type functions in free terms and kernels. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get a priori estimate for solution via unknown coefficient which will be used to study the inverse problem. The inverse problem is reduced to the equivalent integral equation of Volterra type. To show existence unique solution to this equation the Schauder principle is applied. The local existence and uniqueness results are obtained. [ABSTRACT FROM AUTHOR]

A fundamental solution matrix for elliptic systems of the second order with constant leading coefficients is constructed. It is used to obtain an integral representation of functions belonging to the Hölder class in a closed domain with a Lyapunov boundary. In the case of an infinite domain, these functions have power-law asymptotics at infinity. The representation is used to study a mixed-contact boundary value problem for a second-order elliptic system with piecewise constant leading coefficients. The problem is reduced to a system of integral equations that are Fredholm in the domain and singular at its boundary. [ABSTRACT FROM AUTHOR]

This article presents a study on surface gravity wave scattering by a rectangular (box-type) breakwater with thin side plates in the situation of oblique incident waves in deep water. Applying the continuity of fluid pressure and velocity to Havelock’s expansion of velocity potentials, the problem is converted to an integral equation of the Fredholm type, whose solution is the horizontal component of fluid velocity. The integral equation is solved by employing Galerkin’s approximation with polynomials as basis functions multiplied by suitable weight functions. The wave reflection and transmission coefficients are calculated numerically to find the breakwater’s performance on wave scattering. The accuracy of the results is verified through numerical convergence and checking of the energy balance equation. The rectangular breakwater reflects long waves to some extent in water of infinite depth, in contrast to a thin breakwater. The thin plates attached to the rectangular breakwater show a reduction in wave transmission. Furthermore, the attachment of thin plates leads to an increment in horizontal force and a reduction in vertical force. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-point boundary-value problem with a Riemann–Liouville–Caputo fractional derivative of order α ∈ (1 , 2) . We solve this boundary-value problem in a sequence of processes, first using the shooting technique based on the secant iterative method, we convert the boundary-value problem into an initial-value problem, then the initial-value problem is transformed into an equivalent Volterra integral equation with weakly singular kernel. Finally, we find the approximate solution of the resultant equation by using a discretization scheme on a uniform mesh. The method's convergence analysis has been thoroughly established, and it demonstrates that the scheme is first-order convergent. To show the effectiveness of the suggested approach, numerical results are provided. [ABSTRACT FROM AUTHOR]

In this article, we consider the following non-linear singular integral equation y (t) = f (t) + ∫ 0 1 k (t , s) 1 [ y (s) ] α d s in the space of continuous functions on a bounded and closed interval [0, 1] for f ∈ C [ 0 , 1 ] with f > 0 , the kernel k(t, s) is a non-negative continuous function on [ 0 , 1 ] × [ 0 , 1 ] and α > 0 fixed parameter. The non-linearity in the considered equation is singular at the dependent variable y = 0. The existence of a continuous positive solution of this integral equation is well established in the literature. We determine the range of α that guarantees the uniqueness of the continuous positive solution of the integral equation. [ABSTRACT FROM AUTHOR]

We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the -dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem in which an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of this kernel. Next, assuming that there are two solutions and of the stated problem, an equation is formed for the difference of this solution. Further research is being conducted for the difference of solutions of the problem and using the techniques of integral equations estimates. It is shown that, if the unknown kernel can be represented as , then . Thus, the theorem on the uniqueness of the solution of the problem is proved. [ABSTRACT FROM AUTHOR]

The existence and uniqueness theorems of the solution to the boundary-value problem for a parabolic-hyperbolic fractional-order equation with the gluing condition are proved. [ABSTRACT FROM AUTHOR]

An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green's 3rd identity yielding a fixed-point problem for the electric potential gradient and ion concentrations. The integrals are discretized by a combination of midpoint and trapezoid rules, and the resulting algebraic equations are solved by Gummel iteration. Numerical tests for electroneutral and non-electroneutral systems demonstrate the method's 2nd order accuracy and ability to resolve sharp boundary layers. The method is applied to a 1D model of the K + ion channel with a fixed charge density that ensures cation selectivity. In these tests, the proposed integral equation method yields potential and concentration profiles in good agreement with published results. [ABSTRACT FROM AUTHOR]

In this paper, an inverse problem of determining a kernel in a one-dimensional integro-differential time-fractional diffusion equation with initial-boundary and overdetermination conditions is investigated. An auxiliary problem equivalent to the problem is introduced first. By Fourier method this auxilary problem is reduced to equivalent integral equations. Then, using estimates of the Mittag–Leffler function and successive aproximation method, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel which will be used in study of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness results are proven. [ABSTRACT FROM AUTHOR]

This paper considers the inverse problem of determining the time-dependent coefficient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag–Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved. [ABSTRACT FROM AUTHOR]

The fractional Fourier transform (FRFT) is a generalized form of the Fourier transform (FT), and it is another important class of time-frequency analysis tool in signal processing. In this paper, we generalize the three-dimensional (3-D) FRFT to the spherical polar coordinates setting. Then, some properties are obtained associate with the 3-D FRFT such as the inverse transform, spatial shift and multiplication. Moreover, the new convolution theorem and correlation theorem for the 3-D FRFT is studied by these properties, and then Parseval theorem of the 3-D FRFT is also obtained. [ABSTRACT FROM AUTHOR]

The article is dedicated to the analysis of electrodynamic properties of the cylindrical spiral structure. Internal problem for the considered structure in the framework of the thin-wire approximation is reduced to a pair of independent integral Fredholm equations of the first kind, written with respect to even and odd components of the unknown current function. A study of spectral characteristics of the integral operator of the integral equation for various electrical lengths and ratios cylinder radius to its height for a fixed number of turns. Using corresponding systems of projection functions in the framework of the method of moments two independent systems of linear algebraic equations are formed with respect to the unknown expansion coefficients of the even and odd components current function. Certain features of the behavior of eigenfunctions in depending on the wave dimensions and geometrical parameters of the spiral. Features of the frequency dependence of the eigenvalues integral operator were revealed. The conclusion is made about the resonant nature of these dependencies, what makes a cylindrical spiral structure in many respects similar to the previously considered tubular vibrator, a spherical spiral particle and an elliptical frame structure. A hypothesis is put forward about the relationship between the quality factor of resonances of eigenvalues and the emergence of a traveling wave mode in the spiral. Presented results contribute to the formation of an in-depth understanding of emerging in the structure internal processes and serve as a guideline in the construction of approximation models for solving an internal problem. [ABSTRACT FROM AUTHOR]

Integral equations of a number of problems of diffraction of elastic waves on defects, located on the layers interface in plane-parallel layered media, are considered by the method of the over-determined boundary value problem. Solvability conditions for over-determined problems are written in terms of impedance matrices. Various options for finding approximate solutions of integral equations of diffraction problems are considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the fractional (p , q) -Mellin transform of order α , denoted by M p , q α , an appropriate convolution and the inversion formula for M p , q α . We provide several properties of M p , q α including those involving fractional integral and fractional derivative of Riemann–Liouville type. Towards applications, using M p , q α , we solve some integral equations, a fractional (p, q)-difference equation and a Titchmarsh type theorem is proved. [ABSTRACT FROM AUTHOR]

Surface tension plays a central role in the mechanical behavior of soft materials such as gels. Elastocapillary deformation of elastic graded substrates is ubiquitous in soft materials. In this work, the effect of a partially wetting sessile liquid droplet on the elastocapillary deformation of a soft elastic graded substrate is studied. The modulus is assumed to have an exponential form along the thickness direction. By applying the Fourier transformation, a mixed boundary-value problem is reduced into a dual integral equation. The numerical results show that the surface displacement is strongly affected by the inhomogeneity of the material. The study of the wetting properties of gel substrates is essential for both understanding the wetting phenomena of gels and developing gels for applications as soft actuators and sensors that can be used in wearable electronics and soft robotics. [ABSTRACT FROM AUTHOR]

Direct and inverse problems for the equation of forced vibrations of a finite length beam with a variable stiffness coefficient at the lowest term are investigated. The direct problem is the initial–boundary value problem for this equation with boundary conditions in the form of a beam fixed at one end and free at the other. The unknown variable in the inverse problem is a multiplier in the right-hand side, which depends on the space variable . This unknown is determined with respect to the solution of the direct problem by specifying an integral redefinition condition. The uniqueness of the solution of the direct problem is proved by the method of energy estimates. The eigenvalues and eigenfunctions of the corresponding elliptic operator are used to reduce the problems to integral equations. The method of successive approximations is used to prove existence and uniqueness theorems for solutions of these equations. [ABSTRACT FROM AUTHOR]

The paper presents formulations of inverse problems for the Helmholtz equation on finding its right-hand side with a Samarskii–Ionkin-type additional integral condition and the justification of their well-posedness in the Hadamard sense in the class of regular solutions. The uniqueness of solutions of the formulated problems is proved on the basis of integral identities. Solutions of the problem are found in explicit form by the methods of separation of variables and integral equations. [ABSTRACT FROM AUTHOR]

In the paper, we consider a boundary value problem for a third-order mixed differential equation of parabolic-hyperbolic type with a fractional Gerasimov–Caputo operator. Under certain conditions on the data of a problem, applying the methods of the theory of integral equations and the Green function, we prove the unique solvability of the problem. The uniqueness of the solution is proved by the method of the extremum principle, and the existence of this problem is proved by reducing it to a boundary value problem for a fractional order differential equation, as well as to the Volterra integral equation of the second kind. [ABSTRACT FROM AUTHOR]

Our concern is to consider linear integral equations with two delays. A new result on necessary and sufficient conditions for the exponential stability is presented by careful study of the characteristic equation. Explicit expressions of limits of solutions in the critical case where integral equations lose exponential stability are also given by use of an asymptotic formula for solutions. [ABSTRACT FROM AUTHOR]

In this article, we prove some novel fixed-point results for a pair of multivalued dominated mappings obeying a new generalized Nashine–Wardowski–Feng–Liu-type contraction for orbitally lower semi-continuous functions in a complete orbital b-metric space. Furthermore, some new fixed-point theorems for dominated multivalued mappings are established in the scenario of ordered complete orbital b-metric spaces. Some examples are offered to demonstrate the validity of our new results' premise. To demonstrate the applicability of our findings, applications for a system of nonlinear Volterra-type integral equations and fractional differential equations are shown. These results extend the theoretical results of Nashine et al. (Nonlinear Anal., Model. Control 26(3):522–533, 2021). [ABSTRACT FROM AUTHOR]

In this paper, without using neither the compactness nor the uniform convexity, some fixed point theorems are proved by using a binary relation in the setting of a new class of spaces called -partial metric spaces. This class of spaces can be considered the first generalization of metric spaces such that the generated topology is a Hausdorff topology. Our theorems generalize and improve very recent fixed point results in the literature. Finally, we show the existence of a solution for a class of differential equations under new weak conditions. [ABSTRACT FROM AUTHOR]

Inversion of magnetotelluric (MT) responses has been used to explore the electrical conductivity distribution of the Earth's interior. In three-dimensional (3-D) inversions, it is significant to use a good initial model, because final model obtained by most 3-D inversion methods is influenced by the initial model. Although uniform initial models are widely used in 3-D inversions, one-dimensional (1-D) initial models are alternatives, which could more appropriately represent the actual conductivity variations in the Earth's interior. This study presents a two-step 3-D inversion method, especially for marine cases. This inversion method first concentrates on obtaining a 1-D initial model and then inverts for 3-D conductivity structures with it, in both of which the 3-D topography is carefully taken into consideration. This method was tested by synthetic models of different topography variations (depression-shaped, smoothly varying, channel-shaped and square-shaped plateau topography) and of heterogeneous layers with different checkerboard-type anomalies (sharp or smooth lateral conductivity variations) embedded in 1-D model of depth-dependent conductivity. The comparisons were done about obtaining an initial model by the proposed method and that inverted from the corrected responses. The results of 3-D inversions by using the method of this study were also compared to that with different uniform initial models. Results of synthetic tests and comparisons were discussed by using directional information of newly introduced model-vector parameters. The performance and validity of this method was verified. It also revealed that some of the newly introduced model-vector parameters could be used to show the convergence of inversions and help to select inverse model. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem of determining u(x, t) and k(t) functions in the one-dimensional integro-differential heat equation with the nonlocal initial-boundary and over-determination conditions. The unique solvability of the direct problem is rigorously proved using the Fourier method and Schauder principle. To investigate the solvability of the inverse problem, we first consider an auxiliary inverse boundary value problem, which is equivalent to the original one. Then using the Fourier method, the problem is reduced by an equivalent closed system of integral equations with respect to unknown functions. The existence and uniqueness theorem for this system of integral equations is proved by contraction mappings principle. [ABSTRACT FROM AUTHOR]

This paper investigates the existence and stability of random solutions of a class of Hadamard fractional order functional partial integral equations with random effects in Banach spaces. [ABSTRACT FROM AUTHOR]

In this manuscript, a novel general class of contractions, called Jaggi–Suzuki-type hybrid (G-α-ϕ)-contraction, is introduced and some fixed point theorems that cannot be deduced from their akin in metric spaces are proved. The dominance of this family of contractions is that its contractive inequality can be specialized in various manners, depending on multiple parameters. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. Consequently, a number of corollaries that reduce our result to some prominent results in the literature are highlighted and analyzed. Additionally, we examine Ulam-type stability and well-posedness for the new contraction proposed herein. Finally, one of our obtained corollaries is applied to set up unprecedented existence conditions for solution to a class of integral equations. For future aspects of our results, an open problem is noted concerning the discretized population balance model, whose solution may be analyzed using the techniques established herein. [ABSTRACT FROM AUTHOR]

A generalized quadrature method is studied for Volterra integral equations with highly oscillatory kernels. According to the kernel, a two-point quadrature rule is constructed by Lagrange's identity at first. The error of the quadrature formula is presented as well. Then, it is employed to discretize the highly oscillatory equation without the need to compute moment. For the convergence, the asymptotic order as well as the classical order of the quadrature method for equation is analyzed. It is shown that the method has asymptotic order two and converges with order two as step length decreases. Some numerical examples are conducted to test its efficiency. [ABSTRACT FROM AUTHOR]

In this paper, a boundary control problem for a parabolic equation on the interval is considered. In the part of the bound of the given region value of the solution is given and control, which gives us the average value of solution, is founded. By the method of mathematical physics is proved that control exists and the estimate of a minimal time for achieving the given average temperature over some domain is founded. [ABSTRACT FROM AUTHOR]

We study a discrete analogue on the Neumann boundary value problem for elliptic pseudo-differential equation in a quadrant. This approach is based on a special factorization of an elliptic symbol which permits to construct a general solution for a discrete pseudo-differential equation in discrete analogues of Sobolev–Slobodetskii spaces. The discrete Neumann boundary conditions are considered in the paper. Unique solvability of discrete Neumann boundary value problem is proved and a comparison between discrete and continuous solutions is given. [ABSTRACT FROM AUTHOR]

The Kantorovich theory plays an important role in the study of nonlinear equations. It is used to establish the existence of a solution for an equation defined in an abstract space. The solution is usually determined by using an iterative process such as Newton's or its variants. A plethora of convergence results are available based mainly on Lipschitz-like conditions on the derivatives, and the celebrated Kantorovich convergence criterion. But there are even simple real equations for which this criterion is not satisfied. Consequently, the applicability of the theory is limited. The question there arises: is it possible to extend this theory without adding convergence conditions? The answer is, Yes! This is the novelty and motivation for this paper. Other extensions include the determination of better information about the solution, i.e. its uniqueness ball; the ratio of quadratic convergence as well as more precise error analysis. The numerical section contains a Hammerstein-type nonlinear equation and other examples as applications. [ABSTRACT FROM AUTHOR]

In this paper, an inverse problem of determining a time-dependent source coefficient in a one-dimensional time-fractional diffusion equation is investigated with initial-boundary and overdetermination conditions. Before, equivalent auxiliary problem is obtained for direct problem. By Fourier method this auxiliary problem is reduced to equivalent integral equation. Then, using estimates of the Mittag-Leffler function and generalized singular Gronwall inequalities, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient. The inverse problem is reduced to the equivalent integral equation. For solving this equation the contracted mapping principle is applied. The local existence and global uniqueness are proven. Also the stability estimate is obtained. [ABSTRACT FROM AUTHOR]

The Tricomi boundary value problem for a mixed-type equation with a delayed argument of the required function and its derivative is investigated. The problem is uniquely solvable. [ABSTRACT FROM AUTHOR]

M.M. Lavrentiev's linear integral equation arises as a result of a special transformation of a nonlinear coefficient inverse wave sensing problem. The completeness of the set of products of regular harmonic functions and Newtonian potentials supported by a segment is proved. As a corollary, we establish the uniqueness of the solution to M.M. Lavrentiev's equation and a related inverse problem of wave sensing. We present results of an approximate solution of this equation by using parallel calculations. [ABSTRACT FROM AUTHOR]

The electromagnetic wave diffraction from perfectly conducting truncated wedges is considered on the rigorous level in cylindrical coordinates. An analytical regularization method is developed to obtain mathematically accurate problem solutions. The solution method is based on the unknown field representation through the principal value Kontorovich–Lebedev integral and the eigenfunctions series. We analyze the scattering from the semi-infinite truncated wedge, which consists of two non-parallel and non-intersecting perfectly conducting and infinitely thin half-planes, and develop this technique for analysis of more complicated problems of wave diffraction from the truncated wedge of finite length. The problems are reduced to the infinite systems of linear algebraic equations (ISLAE) of the first kind. The convolution type operators and their inverse ones are used to reduce them to the ISLAE of the second kind applied to the analytical regularization procedure. Two versions of the procedure, such as left- and right-sides regularization, are considered. The developed technique is compared with the Wiener–Hopf method. The numerical examples of wave scattering from the truncated wedge, including its well-known geometries as the semi-plane and the slit in the infinite plane, are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear inverse problem for the time-fractional diffusion equation with overdetermination conditions is investigated. Firstly, we studied direct problem. By the Fourier method, this direct problem is reduced to equivalent integral equations. Then, using estimates of the Mittag-Leffler function and generalized singular Gronwall inequalities, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown functions. The inverse problem is reduced to the equivalent system of integral equations. For solving this system, the contracted mapping principle is applied. The local existence and global uniqueness results are proven. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the Moore–Penrose inverse and characteristic matrix of unbounded WCT operators on the Hilbert space L 2 (μ) . In addition, we obtain some applications of the Moore–Penrose inverse of unbounded operators on the Hilbert space H to variational regularization problem. Moreover, some examples are provided to illustrate the applications of our results in linear equations and specially Fredholm integral equations. [ABSTRACT FROM AUTHOR]

The objective of the present research is to establish and prove some new common fuzzy fixed-point theorems for fuzzy set-valued mappings involving Θ-contractions in a complete metric space. For this purpose, a novel integral-type contraction condition is applied to obtain these results. In this way, several useful and classical results have been generalized. Moreover, a concrete example is created to furnish our results. An application to stochastic Volterra integral equations has been given to enhance the validity of our results. [ABSTRACT FROM AUTHOR]

Pfaff equations with continuous coefficients are considered. A specific Cauchy problem for a Pfaff equation is transformed to an equivalent system of integral equations of a special type, which is overdetermined. It is shown that in the case of smooth coefficients the consistency of the system is equivalent to the Frobenius integrability criterion. A theorem on the existence of a solution for the obtained type of integral equations is presented. The solution is found by the Euler polygonal method, which allows one to construct an approximate solution of the Pfaff equation. An analog of Nagumo's theorem on the uniqueness of the solution to the Cauchy problem is also given. [ABSTRACT FROM AUTHOR]

A direct method (self-regularization method) for the numerical solution of a weakly singular integral equation of the first kind on a closed surface is considered. This equation is an integral formulation of the internal and external three-dimensional Dirichlet problems for the Laplace equation if their solutions are sought in the form of a single-layer potential. It is approximated by a system of linear algebraic equations, which is solved numerically. In this case, a new method of averaging the kernel of the integral operator is used. It preserves the conditional correctness of the discretized problem and significantly increases the rate of convergence of its solution to the exact solution of the integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for nonlinear Hammerstein, Volterra–Hammerstein, and Volterra equations in the space of functions of bounded second variation in the sense of Shiba, (Λ p 2 B V [ a , b ]) . [ABSTRACT FROM AUTHOR]